Kinematic Diffusion of Scalar Quantities in Turbulent Velocity Fields

Abstract

From Kraichnan's direct interaction approximation the normal mode equations are set up for a scalar quantity diffusing kinematically under a turbulent velocity field which is statistically homogeneous and stationary. It is demonstrated: (i) that the mean scalar field responds only to the symmetric part of the velocity turbulence tensor; (ii) that the Kraichnan equation describing the normal mode behaviour is a singular nonlinear integral equation; (iii) that for velocity turbulence which is switched on and off infinitely rapidly the normal modes of the mean scalar field decay in time at a rate which is always greater than that obtaining in the absence of the turbulent velocity field. The motivation underlying these calculations is the problem of particle transport in turbulent astrophysical situations such as the interstellar medium. In such cases the effective Reynolds number is normally large compared with unity, so that expansion approximations for small Reynolds number are apparently not completely free of error.